Related papers: A Carousel Property for Compact Convex Sets

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that…

We introduce two sets of invariants for a line bundle at a point: infinitesimal successive minima and asymptotic partial jet separation. They are inspired by the local analogue of Ambro-Ito, and by the jet-theoretic interpretation of the…

We prove that if a compact set E in complex Euclidean space is contained in an arc J, then there is a choice of J whose polynomial hull is the union of J and the polynomial hull of E. This strengthens an earlier result of the author. We…

For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In the present paper it is shown that its convexity implies that it is a $C^1$…

In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

Suppose $G$ is finitely generated group and $\mathcal{C}(G)$ consists of all $\rho:G\to\operatorname{PGL}(n+1,\mathbb{R})$ for which there exists a properly convex set in $\mathbb{R}\mathbb{P}^n$ that is preserved by $\rho(G)$. Then the…

We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not =…

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most…

The paper contains a characterization of compact groups $G\subseteq\GL(V)$, where $V$ is a finite dimensional real vector space, which have the following property \SP{}: the family of convex hulls of $G$-orbits is a semigroup with respect…

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper…

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing…

Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right…

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria…

For convex domains with $C^{1,\epsilon}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the…

We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with…

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n}…

In this paper we present some extensions of the Ailon-Rudnick Theorem, which says that if $f,g\in{\mathbb C}[T]$, then $\gcd(f^n-1,g^m-1)$ is bounded for all $n,m\ge 1$. More precisely, using a uniform bound for the number of torsion points…

We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known constant…